Can units in $M_n(\mathbb{Z})$ be moved to the other side?

Question

Let $M, U_1 \in M_n(\mathbb{Z})$ with $U_1$ a unit (i.e. $\lvert \det(U_1) \rvert=1$). Can I always find another unit $U_2\in M_n(\mathbb{Z})$ such that $U_1 M = M U_2$?

Answer 1

Not if the image of $U_1M$ and that of $M$ differ. I am sure you can think of a concrete example given this hint.

Answer 2

Let $M=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $U_1=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. Then $U_1M=\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, but $MU_2=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix}=\begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix}\neq U_1M$ for all matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$.